Application of the Gauss' method to the stellar three body problem (Q1390806)

The stellar three-body problem with two close stars and the third one located far away from two others is considered. It is supposed that the orbit of the third star is the Keplerian one. The Gauss' method which consists in replacing a gravitational attraction of the third star by the gravitational attraction of an infinitesimal non-homogeneous elliptic ring. Until now the analytic results were related with an elliptic ring of a small eccentricity. However, in general this is not valid when applied to the stellar problem, since the outer orbit is not necessarily almost circular. In the article an analytic method is developed to obtain not potential but the force vector for any value of eccentricity.